Unified Analytic Solution of Polynomial Equations in Non-Commutative Clifford Algebraic Closure: A Comprehensive Treatment with Complete Mathematical Foundations

Unified Analytic Solution of Polynomial Equations in Non-Commutative Clifford Algebraic Closure: A Comprehensive Treatment with Complete Mathematical Foundations | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 24 October 2025 V1 Latest version Share on Unified Analytic Solution of Polynomial Equations in Non-Commutative Clifford Algebraic Closure: A Comprehensive Treatment with Complete Mathematical Foundations Authors : Dongqi Liu 0009-0006-4018-9292 and shifa liu 0009-0003-6570-2812 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176132860.09641042/v1 159 views 116 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract 本文提出了求解多项式方程的微分代数和向量代数框架对非交换克利福德代数设置的全面扩展。我们通过引入一个严格定义的路径排序算子 P，直接面对非交换性带来的基本障碍，该算子 P 投射到对称子空间上，同时保留基本代数结构。我们通过构造非交换克利福德代数闭包 V nc Cl，为求解在非交换克利福德代数 Cl（V， Q） 中定义的多元多项式方程组建立了完整的理论基础。求解方法为此类系统的根提供了显式分析表达式，采用统一形式：非交换牛顿恒等式和收敛分析。开发的算法通过全面的误差分析实现了 O（n 3 m 3 2 3m） 复杂度。数值验证在各种测试用例中展示了 10 −13 残差的精度。这项工作与经典的不可能性结果相协调，同时证明显式解析解存在于适当扩展的代数结构中，这些代数结构在非交换环境中结合了微分和几何运算。 Supplementary Material File (noncommutative cliford (1).pdf) Download 432.67 KB Information & Authors Information Version history V1 Version 1 24 October 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords algebraic closure complete mathematical foundations differential algebra explicit solution galois theory geometric computation machine learning multivariate polynomial systems non-commutative clifford algebra numerical analysis path-ordering Authors Affiliations Dongqi Liu 0009-0006-4018-9292 View all articles by this author shifa liu 0009-0003-6570-2812 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 159 views 116 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Dongqi Liu, shifa liu. Unified Analytic Solution of Polynomial Equations in Non-Commutative Clifford Algebraic Closure: A Comprehensive Treatment with Complete Mathematical Foundations. Authorea . 24 October 2025. DOI: https://doi.org/10.22541/au.176132860.09641042/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . Format Please select one from the list RIS (ProCite, Reference Manager) EndNote BibTex Medlars RefWorks Direct import Tips for downloading citations document.getElementById('citMgrHelpLink').addEventListener('click', function() { popupHelp(this.href); return false; }); $(".js__slcInclude").on("change", function(e){ if ($(this).val() == 'refworks') $('#direct').prop("checked", false); $('#direct').prop("disabled", ($(this).val() == 'refworks')); }); View Options View options PDF View PDF Figures Tables Media Share Share Share article link Copy Link Copied! Copying failed. Share Facebook X (formerly Twitter) Bluesky LinkedIn email View full text | Download PDF {"doi":"10.22541/au.176132860.09641042/v1","type":"Article"} Now Reading: Share Figures Tables Close figure viewer Back to article Figure title goes here Change zoom level Go to figure location within the article Download figure Toggle share panel Toggle share panel Share Toggle information panel Toggle information panel Go to previous graphic Go to next graphic Go to previous table Go to next table All figures All tables View all material View all material xrefBack.goTo xrefBack.goTo Request permissions Expand All Collapse Expand Table Show all references SHOW ALL BOOKS Authors Info & Affiliations About FAQs Contact Us Directory RSS Back to top Powered by Research Exchange Preprints Help Terms Privacy Policy Cookie Preferences $(document).ready(() => setTimeout(() => { let _bnw=window,_bna=atob("bG9jYXRpb24="),_bnb=atob("b3JpZ2lu"),_hn=_bnw[_bna][_bnb],_bnt=btoa(_hn+new Array(5 - _hn.length % 4).join(" ")); $.get("/resource/lodash?t="+_bnt); },4000)); (function(){function c(){var b=a.contentDocument||a.contentWindow.document;if(b){var d=b.createElement('script');d.innerHTML="window.__CF$cv$params={r:'9ffd2c8d9e0bdfa9',t:'MTc3OTQ2NzA5Nw=='};var a=document.createElement('script');a.src='/cdn-cgi/challenge-platform/scripts/jsd/main.js';document.getElementsByTagName('head')[0].appendChild(a);";b.getElementsByTagName('head')[0].appendChild(d)}}if(document.body){var a=document.createElement('iframe');a.height=1;a.width=1;a.style.position='absolute';a.style.top=0;a.style.left=0;a.style.border='none';a.style.visibility='hidden';document.body.appendChild(a);if('loading'!==document.readyState)c();else if(window.addEventListener)document.addEventListener('DOMContentLoaded',c);else{var e=document.onreadystatechange||function(){};document.onreadystatechange=function(b){e(b);'loading'!==document.readyState&&(document.onreadystatechange=e,c())}}}})();